Question

m∠nqp = 45
m∠qpn = 53
m∠pnq = 82
rn =
pq =
nm = √

Explanation:

Step1: Confirm $\angle PNQ$

Sum angles in $\triangle PNQ$:
$$m\angle PNQ = 180^\circ - 45^\circ - 53^\circ = 82^\circ$$

Step2: Calculate $RN$

Use $\triangle PRN$:
$$RN = 24\sqrt{2} \times \sin45^\circ = 24\sqrt{2} \times \frac{\sqrt{2}}{2} = 24$$

Step3: Calculate $PR$

Use $\triangle PRN$:
$$PR = 24\sqrt{2} \times \cos45^\circ = 24\sqrt{2} \times \frac{\sqrt{2}}{2} = 24$$

Step4: Calculate $PQ$

Sum $PR$ and $RQ$:
$$PQ = 24 + 18 + 24 = 66$$

Step5: Calculate $NM$

Use $\triangle SNM$:
$$NM = \sqrt{16^2 + 12^2} = \sqrt{256+144} = \sqrt{400} = 20\sqrt{2}$$

Answer:

$m\angle NQP = 45$
$m\angle QPN = 53$
$m\angle PNQ = 82$
$RN = 24$
$PQ = 66$
$NM = 20\sqrt{2}$